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Lévy constants of transcendental numbers. (English) Zbl 1179.11020
For a real irrational number \(\alpha\), let \(q_n(\alpha)\) denote the denominator of the \(n\)’th convergent to \(\alpha\) coming from the simple continued fraction expansion of \(\alpha\). The number \(\alpha\) is said to have Lévy contant \(\beta(\alpha)\) if the limit \[ \beta(\alpha) = \lim_{n \rightarrow \infty} {{\log q_n(\alpha)} \over n} \] exists. A classical result of P. Lévy [Compos. Math. 3, 286–303 (1936; Zbl 0014.26803)] states that this constant exists and is equal to \(\pi^2/(12 \log 2)\) for almost every real number \(\alpha\) with respect to the Lebesgue measure. By considering the Golden Ratio \((1+\sqrt{5})/2\), it can easily be seen that whenever \(\beta(\alpha)\) exists, it is at least equal to \(\log((1+\sqrt{5})/2)\).
In the present paper, the author proves that for any number \(\gamma \geq \log((1+\sqrt{5})/2)\), there are uncountable many pairwise inequivalent transcendental numbers \(\alpha\), such that \(\beta(\alpha) = \gamma\). He then goes on to prove that this also holds true when we require the transcendental numbers to be \(U_2\)-numbers in Mahler’s classification of transcendental numbers.
MSC:
11J81 Transcendence (general theory)
11J82 Measures of irrationality and of transcendence
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